= Zero-state response
{title2=$Y=HX$}
The zero-state response is the output produced by an applied input when the system starts with zero internal state. For a <linear time-invariant system> it separates the effect of the input from the <zero-input response> caused by an initially stored state. A zero initial output alone need not mean zero internal state.
For a causal <linear time-invariant system> with <impulse response> $h$, the zero-state response is the <convolution>
$$
y_{\rm zs}(t)=(h*u)(t)=\int_0^t h(t-s)u(s)\,ds.
$$
An instantaneous feedthrough term, if present, is represented by a <Dirac delta> term in $h$. Equivalently, for constant matrices in $\dot x=Ax+Bu$, $y=Cx+Du$ with $x(0)=0$,
$$
x(t)=\int_0^t e^{A(t-s)}Bu(s)\,ds,\qquad
y_{\rm zs}(t)=C\int_0^t e^{A(t-s)}Bu(s)\,ds+Du(t).
$$
For nonzero $x(0)=x_0$, <linearity> adds the <zero-input response> $Ce^{At}x_0$. Here the <matrix exponential> propagates the state left by the earlier input.
For example, $\tau\dot y+y=u$ with $\tau>0$, $y(0)=0$ and a constant input $u(t)=A$ for $t\geq0$ has the <step response> $y_{\rm zs}=A(1-e^{-t/\tau})$. At one <time constant> it has reached about $63.2\%$ of its final value. If the initial value instead equals $y_0$, the total output is $A(1-e^{-t/\tau})+y_0e^{-t/\tau}$. A charged <capacitor> in an <RC circuit> provides a direct physical example: discharge it before repeating the same input step to isolate the zero-state response, and compare with a run from a known initial <electric potential difference>.<State and Memory by Anant Agarwal and Jeffrey Lang>[PDF pages 2–6, capacitor memory and separate zero-state/zero-input responses]{ref}
With compatible transforms, the zero-state input-output relation is $Y=HU$, where $H$ is the <transfer function>. For a nonzero initial state, a unilateral <Laplace transform> includes additional initial-condition terms. Those terms cannot be absorbed into an input-independent <transfer function> multiplying $U$.
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